ds.dynamical systems - When is an Anosov diffeomorphism mixing?

I don't have a proper answer to your main question beyond pointing to the list of equivalent properties in Pesin's book (your first reference), which you've obviously seen already. However, I'll point out that in Ruelle's paper (your third reference), the first main theorem (on page 3), which contains a statement on exponential decay of correlations, is proved under the hypothesis that the unstable manifolds are dense in the attractor. As in Pesin's book, that will imply mixing (on the attractor), so there's no mystery in the coexistence of the result stated by Ruelle and the open problem stated by Pesin.

As far as I know the best that you can say in general is that Anosov diffeos are mixing on the non-wandering set, or on the closure of an unstable manifold. So an equivalent question is, "Under what circumstances is every point non-wandering for an Anosov diffeo?" (Or, "Under what circumstances are unstable manifolds dense?")